Textbook Question
{Use of Tech} Equations of tangent lines
b. Use a graphing utility to graph the curve and the tangent line on the same set of axes.
y = e^x; a = ln 3
2
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Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 3, Problem 60
Calculate the derivative of the following functions.
y = √x+√x+√x
Verified step by step guidance1
Step 1: Recognize that the function y = \(\sqrt{x}\) + \(\sqrt{x}\) + \(\sqrt{x}\) can be simplified to y = 3\(\sqrt{x}\).
Step 2: Recall the power rule for derivatives, which states that if y = x^n, then the derivative y' = nx^{n-1}.
Step 3: Rewrite \(\sqrt{x}\) as x^{1/2} to apply the power rule. Therefore, y = 3x^{1/2}.
Step 4: Differentiate y = 3x^{1/2} using the power rule. The derivative of x^{1/2} is (1/2)x^{-1/2}.
Step 5: Multiply the derivative of x^{1/2} by the constant 3 to get the final derivative: y' = 3 * (1/2)x^{-1/2}.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative
The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In calculus, the derivative is often denoted as f'(x) or dy/dx, and it provides critical information about the function's behavior, such as its slope and concavity.
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Derivatives
Chain Rule
The chain rule is a fundamental technique for finding the derivative of composite functions. It states that if a function y is composed of two functions u and x (i.e., y = f(u) and u = g(x)), then the derivative of y with respect to x can be found by multiplying the derivative of f with respect to u by the derivative of g with respect to x. This rule is essential when differentiating functions that involve nested expressions.
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Power Rule
The power rule is a basic rule for differentiating functions of the form f(x) = x^n, where n is a real number. According to this rule, the derivative f'(x) is given by n*x^(n-1). This rule simplifies the process of differentiation for polynomial and root functions, making it easier to compute derivatives quickly and efficiently.
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Related Practice
Textbook Question
Determining the unknown constant Let f(x) = {2x² if x≤1 ax-2 if x>1. Determine a value of a (if possible) for which f' is continuous at x=1.
Textbook Question
Calculate the derivative of the following functions.
y = (f(g(x^m)))^n, where f and g are differentiable for all real numbers and m and n are constants
Textbook Question
{Use of Tech} Equations of tangent lines
Find an equation of the line tangent to the given curve at a.
y = −3x2 + 2; a=1
1
views
Textbook Question
{Use of Tech} Equations of tangent lines
Find an equation of the line tangent to the given curve at a.
y = ex; a = ln 3
2
views
Textbook Question
{Use of Tech} Equations of tangent lines
b. Use a graphing utility to graph the curve and the tangent line on the same set of axes.
y = −3x²+2; a=1
1
views